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Nevanlinna Theory
In the mathematical field of complex analysis, Nevanlinna theory is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl called it "one of the few great mathematical events of (the twentieth) century." The theory describes the asymptotic distribution of solutions of the equation ''f''(''z'') = ''a'', as ''a'' varies. A fundamental tool is the Nevanlinna characteristic ''T''(''r'', ''f'') which measures the rate of growth of a meromorphic function. Other main contributors in the first half of the 20th century were Lars Ahlfors, André Bloch, Henri Cartan, Edward Collingwood, Otto Frostman, Frithiof Nevanlinna, Henrik Selberg, Tatsujiro Shimizu, Oswald Teichmüller, and Georges Valiron. In its original form, Nevanlinna theory deals with meromorphic functions of one complex variable defined in a disc , ''z'', ≤ ''R'' or in the whole complex plane (''R'' = ∞). Subsequent generalizations extended Nevanlinna theory to algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiregular Map
In the mathematical field of analysis, quasiregular maps are a class of continuous maps between Euclidean spaces R''n'' of the same dimension or, more generally, between Riemannian manifolds of the same dimension, which share some of the basic properties with holomorphic functions of one complex variable. Motivation The theory of holomorphic (= analytic) functions of one complex variable is one of the most beautiful and most useful parts of the whole mathematics. One drawback of this theory is that it deals only with maps between two-dimensional spaces (Riemann surfaces). The theory of functions of several complex variables has a different character, mainly because analytic functions of several variables are not conformal. Conformal maps can be defined between Euclidean spaces of arbitrary dimension, but when the dimension is greater than 2, this class of maps is very small: it consists of Möbius transformations only. This is a theorem of Joseph Liouville; relaxing the smoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ahlfors Theory
Ahlfors theory is a mathematical theory invented by Lars Ahlfors as a geometric counterpart of the Nevanlinna theory. Ahlfors was awarded one of the two very first Fields Medals for this theory in 1936. It can be considered as a generalization of the basic properties of covering maps to the maps which are "almost coverings" in some well defined sense. It applies to bordered Riemann surfaces equipped with conformal Riemannian metrics. Preliminaries A ''bordered Riemann surface'' ''X'' can be defined as a region on a compact Riemann surface whose boundary ∂''X'' consists of finitely many disjoint Jordan curves. In most applications these curves are piecewise analytic, but there is some explicit minimal regularity condition on these curves which is necessary to make the theory work; it is called the ''Ahlfors regularity''. A ''conformal Riemannian metric'' is defined by a length element ''ds'' which is expressed in conformal local coordinates ''z'' as ''ds'' = ''ρ''(''z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Bonnet Theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a triangle on a plane, the sum of its angles is 180 degrees. The Gauss–Bonnet theorem extends this to more complicated shapes and curved surfaces, connecting the local and global geometries. The theorem is named after Carl Friedrich Gauss, who developed a version but never published it, and Pierre Ossian Bonnet, who published a special case in 1848. Statement Suppose is a compact two-dimensional Riemannian manifold with boundary . Let be the Gaussian curvature of , and let be the geodesic curvature of . Then :\int_M K\,dA+\int_k_g\,ds=2\pi\chi(M), \, where is the element of area of the surface, and is the line element along the boundary of . Here, is the Euler characteristic of . If the boundary is piecewise smooth, then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula \frac where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f''; that is, the infinitesimal absolute change in ''f,'' namely f', scaled by the current value of ''f.'' When ''f'' is a function ''f''(''x'') of a real variable ''x'', and takes real, strictly positive values, this is equal to the derivative of ln(''f''), or the natural logarithm of ''f''. This follows directly from the chain rule: \frac\ln f(x) = \frac \frac Basic properties Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does ''not'' take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' . So for positive-real-valued functions, the logarithmic d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard's Theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. The theorems Little Picard Theorem: If a function f: \mathbb \to\mathbb is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted by λ, and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If ''f'' omits two values, then the composition of ''f'' with the inverse of the modular function maps the plane into the unit disc which implies that ''f'' is constant by Liouville's theorem. This theorem is a significant strengthening of Liouville's theorem which states that t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to the set of complex numbers with positive imaginary part: :\mathcal \equiv \ ~. The term arises from a common visualization of the complex number as the point in the plane endowed with Cartesian coordinates. When the axis is oriented vertically, the "upper half-plane" corresponds to the region above the axis and thus complex numbers for which > 0. It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by 0. Proposition: Let ''A'' and ''B'' be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes ''A'' to ''B''. :Proof: First shift the center of ''A'' to (0,0). Then take λ = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Type (mathematics)
In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a function is of bounded type in a region \Omega if and only if f is analytic on \Omega and \log^+, f(z), has a harmonic majorant on \Omega, where \log^+(x)=\max ,\log(x)/math>. Being the ratio of two bounded analytic functions is a sufficient condition for a function to be of bounded type (defined in terms of a harmonic majorant), and if \Omega is simply connected the condition is also necessary. The class of all such f on \Omega is commonly denoted N(\Omega) and is sometimes called the '' Nevanlinna class'' for \Omega. The Nevanlinna class includes all the Hardy classes. Functions of bounded type are not necessarily bounded, nor do they have a property called "type" which is bounded. The reason for the name is probably that when defined on a disc, the Nevanlinna characteristic ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the variables may be taken in any field ''L'' containing ''K''. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is ''L''. The set of rational functions over a field ''K'' is a field, the field of fractions of the ring of the polynomial functions over ''K''. Definitions A function f(x) is called a rational function if and only if it can be written in the form : f(x) = \frac where P\, and Q\, are polynomial functions of x\, and Q\, is not the zero function. The domain of f\, is the set of all valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entire Function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function has a root at , then , taking the limit value at , is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function. A transcendental entire function is an entire function that is not a polynomial. Properties Every entire function can be represented as a power series f(z) = \sum_^\infty a_n z^n that converges everywhere in the complex plane, hence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value \infty for infinity. With the Riemann model, the point \infty is near to very large numbers, just as the point 0 is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=\infty well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical example of a Riemann surface, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jensen's Formula
In the mathematical field known as complex analysis, Jensen's formula, introduced by , relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle. It forms an important statement in the study of entire functions. The statement Suppose that ''ƒ'' is an analytic function in a region in the complex plane which contains the closed disk D of radius ''r'' about the origin, ''a''1, ''a''2, ..., ''a''''n'' are the zeros of ''ƒ'' in the interior of D (repeated according to their respective multiplicity), and that ''ƒ''(z) ≠ 0 for all z \in\partialD, as well as ''ƒ''(0) ≠ 0. Jensen's formula states that :\log , f(0), = \sum_^n \log \left( \frac\right) + \frac \int_0^ \log, f(re^), \, d\theta. This formula establishes a connection between the moduli of the zeros of the function ''ƒ'' inside the disk D and the average of log , ''f''(''z''), on the boundary circle , ''z'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
